Hyperbola Showdown: Comparing Equations & Features

Oct 30, 2025 by SLV Team 51 views

Hey guys! Let's dive into some math and compare two interesting hyperbolas. We're going to break down the graphs of $\frac{x^2}{3^2}-\frac{y^2}{4^2}=1$ and $\frac{y^2}{3^2}-\frac{x^2}{4^2}=1$ to see how they stack up. This isn't just about equations; we'll look at the cool features like foci, axes, and directrices to figure out what's the same and what's different. Understanding hyperbolas can be tricky, but I'll try to make it as easy and fun as possible! So, grab your pencils (or your favorite digital drawing tools), and let's get started. This is going to be a fun journey where we'll explore different components of hyperbolas! Ready?

Unveiling the Hyperbolas: Equations & Orientations

First things first, let's talk about the equations themselves. You've got $\frac{x^2}{3^2}-\frac{y^2}{4^2}=1$ and $\frac{y^2}{3^2}-\frac{x^2}{4^2}=1$ . The key difference here is the order of the $x^2$ and $y^2$ terms. This subtle change has a big impact on the graphs! Remember, that these equations represent hyperbolas, which are special curves formed by two symmetrical branches. Hyperbolas are fascinating mathematical objects that appear in various applications, from physics to architecture. Understanding their properties is crucial for solving problems related to these applications. These two specific equations are great examples to illustrate the key features. Understanding these is essential for our comparison.

The first equation, $\frac{x^2}{3^2}-\frac{y^2}{4^2}=1$ , represents a hyperbola that opens horizontally. This means the branches of the hyperbola stretch out to the left and right. Think of it like a sideways smile. The $x^2$ term comes first and is positive, that's your clue. It centers on the origin (0,0). The values 3 and 4 are important for determining the size and shape. The value 3 is related to the distance from the center to the vertices along the x-axis, and 4 is related to the distance to the co-vertices along the y-axis. The vertices are the points where the hyperbola intersects its transverse axis. The co-vertices are the points that help define the rectangle that helps to make the asymptotes. The asymptote acts like a guide.

On the other hand, the second equation, $\frac{y^2}{3^2}-\frac{x^2}{4^2}=1$ , represents a hyperbola that opens vertically. In this case, the $y^2$ term is positive, indicating that the branches extend upwards and downwards. Imagine it as a sad face. The hyperbola also centers on the origin. Now the 3 and 4 are also swapped from the first. The 3 is related to the distance from the center to the vertices along the y-axis, and 4 is related to the distance to the co-vertices along the x-axis. The vertices are now on the y-axis. These simple observations give us some insights to the key differences. These orientations are vital to understand their properties.

To really get it, let's plot these bad boys. Graphing the equations will help us to visualize their different orientations. You'll see one hugging the x-axis and the other hugging the y-axis. The shape and orientation are crucial to understand the properties. The graph also shows us the difference between the vertices, and how the asymptotes differ.

Key Takeaway:

The first hyperbola opens horizontally.
The second hyperbola opens vertically.

Focus on the Foci: Location, Location, Location!

Alright, let's move on to the foci. The foci (plural of focus) are two special points inside each hyperbola. They're like the heart of the curve, playing a key role in defining its shape. The distance from any point on the hyperbola to the two foci has a constant difference. The foci are crucial in understanding how light or other signals interact with these curves, which is why we must study them. The location of the foci is related to the values in the denominators of the equation.

For any hyperbola, the distance from the center to each focus is denoted as 'c'. We calculate 'c' using the equation $c^2 = a^2 + b^2$ , where $a$ and $b$ are the values under the $x^2$ and $y^2$ terms in the equation. In both cases, the values of $a$ and $b$ are going to be 3 and 4, respectively. This gives us $c^2 = 3^2 + 4^2 = 9 + 16 = 25$ . Therefore, $c = 5$ .

Now, for $\frac{x^2}{3^2}-\frac{y^2}{4^2}=1$ , the foci are located at (-5, 0) and (5, 0). This is because the hyperbola opens horizontally, and the foci lie on the transverse axis (the x-axis in this case).

For $\frac{y^2}{3^2}-\frac{x^2}{4^2}=1$ , the foci are located at (0, -5) and (0, 5). Here, the hyperbola opens vertically, so the foci are on the y-axis.

So, do the foci of both graphs lie at the same points? Nope! The foci are at different locations depending on which axis the hyperbola opens along. One lies on the x-axis, and the other lies on the y-axis.

Key Takeaway:

Foci locations differ due to the different orientations.

Transverse Axes & Their Lengths

Let's talk about the transverse axis. This is the line segment that passes through the two vertices and the center of the hyperbola. It's the axis along which the hyperbola opens. The length of the transverse axis is a key characteristic of the hyperbola.

For the hyperbola $\frac{x^2}{3^2}-\frac{y^2}{4^2}=1$ , the transverse axis lies along the x-axis. The vertices are at (-3, 0) and (3, 0). The length of the transverse axis is the distance between the vertices, which is $2a = 2 imes 3 = 6$ .

For the hyperbola $\frac{y^2}{3^2}-\frac{x^2}{4^2}=1$ , the transverse axis lies along the y-axis. The vertices are at (0, -3) and (0, 3). The length of the transverse axis is again $2a = 2 imes 3 = 6$ .

So, do the lengths of the transverse axes of both graphs match up? Yes! Even though the orientation differs, the length of the transverse axis is the same because the value of 'a' (which determines the length) is the same for both hyperbolas.

Key Takeaway:

The lengths of both transverse axes are the same.

Directrices: Guiding the Curves

Finally, let's explore the directrices. The directrices are lines that are associated with the hyperbola, and they help define its shape. For each focus, there is a corresponding directrix. The distance from any point on the hyperbola to a focus is a constant multiple of its distance to the corresponding directrix.

The distance from the center to each directrix is given by $a^2/c$ . The value of $a = 3$ , and we know that $c = 5$ .

For the hyperbola $\frac{x^2}{3^2}-\frac{y^2}{4^2}=1$ , the directrices are vertical lines at $x = -a^2/c = -9/5$ and $x = a^2/c = 9/5$ . These lines are parallel to the y-axis.

For the hyperbola $\frac{y^2}{3^2}-\frac{x^2}{4^2}=1$ , the directrices are horizontal lines at $y = -a^2/c = -9/5$ and $y = a^2/c = 9/5$ . These lines are parallel to the x-axis.

Do the directrices of the graphs have the same locations? Not quite! The equations for the directrices are different because of the orientation of each hyperbola. The directrices' positions depend on the orientation, and so their location changes. However, both hyperbolas will have directrices parallel to the other axis.

Key Takeaway:

The directrices are located differently, but they are both at a distance $a^2/c$ .

Conclusion: Putting it All Together

So, there you have it, guys! We have explored the graphs of $\frac{x^2}{3^2}-\frac{y^2}{4^2}=1$ and $\frac{y^2}{3^2}-\frac{x^2}{4^2}=1$ . They're both hyperbolas, but they have distinct differences. Let's recap what we've discovered:

Foci: The foci are not the same. The locations depend on the orientation.
Transverse Axes: The lengths are the same. It's determined by the same values in the equations.
Directrices: The directrices are not located in the same place. Their positions are dependent on the orientation.

So, back to the original question! The true statement is: The lengths of both transverse axes are the same.

I hope you enjoyed this journey into the world of hyperbolas! Keep practicing, and you'll become a hyperbola expert in no time. If you have any questions, feel free to ask! Happy math-ing! Keep up the great work!